This is an exercise in Riemannian geometry. In this note I Taylor-expand the Riemannian metric in a normal neighborhood around a point in a Riemannian manifold . What I have done is nothing except providing a few more terms than most standard textbooks. Hopefully the computation is correct. (I haven’t checked against other sources. )
Theorem 1 In a normal coordinates neighborhood of , the Taylor series of around is given by
Proof: In normal coordinates, fix for the moment and let be a radial geodesic, then is a Jacobi field on , where . (This can be seen by observing that is the variational vector field of the -family of geodesics , with a slight abuse of notations. ) Let .
We will use to denote a covariant derivative w.r.t. and define , then implies is a symmetric operator. Differentiating implies are also symmetric. Also, the Jacobi equation simply becomes
We compute (a way to test the correctness of the following computation is to note that each carries two derivatives)
(Actually to compute we don’t have to compute all the terms of : we can omit the term containing any ). It follows that
All the above expression are evaluated at , noting that . So we have (all repeated indices are summed over):
We thus conclude that
Here is the radial distance from .